Why High-Order Global Polynomials Fail in RD

Why the Shape of the Fit Matters

In a regression discontinuity design, the causal contrast is local: it concerns the difference between the outcome limits just below and just above the cutoff. A regression curve is a way to estimate those limits from observed scores. The curve’s behavior far from the cutoff is not itself the target, but a global polynomial fit uses observations across the score range to determine its shape—including its value at the cutoff.

A high-degree polynomial can follow complicated patterns in the observed data. That flexibility is not always useful. In finite samples, a global curve may bend sharply near the cutoff, oscillate between observations, or respond strongly to a few scores at the edge of the data. Its estimated discontinuity can then reflect the polynomial’s global shape as much as the data immediately around the threshold. Local polynomial estimation instead fits a simpler curve using observations within a chosen neighborhood of the cutoff, with nearer observations typically receiving more weight.

Definition: Let \(X_i\) be the running variable, with cutoff normalized to zero, and let \(Y_i\) be the outcome. A degree-\(p\) global polynomial RD fit estimates separate polynomials on the two sides using observations across the chosen score range:

Y_i=\sum_{j=0}^{p}\beta_j^-X_i^j\,1(X_i<0) +\sum_{j=0}^{p}\beta_j^+X_i^j\,1(X_i\geq 0)+\varepsilon_i.

The fitted discontinuity at the cutoff is \(\hat{\tau}_{global}=\hat{\beta}_0^+-\hat{\beta}_0^-\). The superscripts denote the below-cutoff and at-or-above-cutoff sides. In a sharp RD design, the target is the difference between the corresponding outcome limits; interpreting it causally also requires the RD identification assumptions.

The intercepts are the fitted limits because all terms involving \(X_i\) vanish at zero. A global polynomial fit is not the same as a local polynomial fit of the same degree: the global fit uses the full specified score range, while the local fit restricts attention to a neighborhood and may weight observations by their distance from the cutoff. Thus, “degree \(p\)” alone does not describe how an RD curve is estimated.

Why High Degrees Can Mislead

The central concern is not that every high-degree global polynomial must fail. It is that a high degree can create substantial instability without making the cutoff estimate more credible. Several features of RD data make this especially consequential.

Global shape restrictions. One polynomial must describe the outcome pattern across the whole selected range on each side. If the true relationship changes shape, a global fit may compromise across distant regions in a way that gives a poor approximation near zero.
Oscillation and boundary behavior. High-degree polynomials can bend sharply, particularly near the ends of the range used for fitting. A cutoff is a boundary of each side-specific sample, so these bends can affect the estimated intercept directly. A curve that looks smooth overall may still turn rapidly near the threshold.
Influential observations and sparse support. Scores far from zero can have strong influence on a global fit, especially when they lie near an edge or are unusual in outcome. If there are few observations in a tail, the fit there can be driven by a small number of points, yet still shape the curve evaluated at the cutoff.
Numerical and statistical instability. Powers such as \(X_i^2,\ldots,X_i^p\) can be highly correlated over a limited score range. This can make individual coefficients imprecise and fitted values sensitive to small changes in the data. Rescaling scores can help numerical computation, but it does not make an unsuitable global shape appropriate.

These problems are related but distinct. Oscillation describes the shape of the fitted curve; sensitivity describes how much the fit changes under data or specification changes. A visually smooth curve can still have an unstable intercept, and a large estimated discontinuity does not by itself reveal whether it is driven by a sharp local jump or by how the polynomial bends toward the cutoff.

Key distinction: A high-order global polynomial uses flexibility across the entire fitting range to estimate a local RD contrast. A local polynomial uses a restricted neighborhood to estimate that contrast. Local estimation does not eliminate the need to choose a suitable neighborhood or assess uncertainty, but it aligns the fitting task more directly with the cutoff estimand.

How One Observation Can Affect the Cutoff

The sensitivity of a global polynomial estimate can be seen directly from least squares. For a given side of the cutoff, write the regression in matrix form as \(Y=\mathbf{X}\boldsymbol{\beta}+\boldsymbol{\varepsilon}\), where the columns of \(\mathbf{X}\) contain the polynomial powers. If \(\mathbf{X}\) has full column rank, the least-squares estimate is \(\hat{\boldsymbol{\beta}}=(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'Y\). Suppose one observed outcome, \(Y_k\), changes by \(\delta\), with all scores held fixed. Let \(\mathbf{x}_k\) be the corresponding design row, written as a column vector. Then:

\begin{aligned} \hat{\boldsymbol{\beta}}_{\text{new}}-\hat{\boldsymbol{\beta}}_{\text{old}} &=(\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'(\delta\mathbf{e}_k)\\ &=\delta(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x}_k, \end{aligned}

where \(\mathbf{e}_k\) is zero except for a one in position \(k\). The fitted value at the cutoff is the intercept, selected by a vector \(\mathbf{e}_0\) that is one in the intercept position and zero elsewhere. Therefore its change on this side is:

\Delta\hat{\beta}_0 =\delta\,\mathbf{e}_0'(\mathbf{X}'\mathbf{X})^{-1}\mathbf{x}_k.

For an RD discontinuity, the change in the estimated contrast is the change in the right-side intercept minus the change in the left-side intercept. The formula does not say that every distant observation has a large effect. It shows why a high-degree global fit can be sensitive: the effect depends on the design and on how the observation’s score maps through the polynomial basis into the estimated intercept. With more polynomial terms, that mapping can be unstable, particularly at the boundary.

Worked Comparisons With Simulated Score Distributions

The following examples use hypothetical simulated data, not real studies. In each, scores are centered at the cutoff, so \(X=0\) is the threshold. The regression outputs are illustrative summaries of fits to the stated simulated samples. They demonstrate possible failure modes, not universal numerical outcomes.

Worked Example: A Global Fit Changes the Estimated Jump

Suppose a simulated sharp RD sample contains 1,200 units with scores from \(-10\) to \(10\), and a true outcome jump of 1 outcome unit at zero. Scores are reasonably common near the cutoff, and the conditional outcome patterns are nearly linear for \(-2\leq X<0\) and \(0\leq X\leq 2\). A local linear fit using scores within two units estimates cutoff limits of 14.0 below and 15.0 above.

The local linear estimate of the jump is:

\hat{\tau}_{local}=15.0-14.0=1.0\text{ outcome unit}.

Now fit separate quartic polynomials globally over the full score range. Suppose the fitted intercepts are 13.2 below and 16.8 above. The global estimate is:

\hat{\tau}_{global}=16.8-13.2=3.6\text{ outcome units}.

The two estimates differ by \(3.6-1.0=2.6\) outcome units. In this simulation, the global fit’s higher degree does not make its cutoff estimate more accurate: it imposes curves across the entire range, and their extrapolated-to-the-boundary intercepts differ substantially from the local limits. The comparison is a warning to examine the fitted curves and sensitivity, not a rule that the local estimate must always be correct.

Worked Example: A Distant Score Alters a Global Estimate

Consider a second hypothetical simulation with scores on both sides of zero and sparse observations near the far ends of the range. A separate degree-five global fit gives a left intercept of 22.4 and a right intercept of 24.8. Its estimated discontinuity is:

\hat{\tau}_{global}=24.8-22.4=2.4\text{ outcome units}.

One observation has a score of \(X=9.6\), far outside the neighborhood used for a local fit, and an outcome unusually high relative to nearby observations. In a sensitivity calculation, increase that observation’s outcome by 8 units, keeping all scores and other outcomes unchanged. The refitted degree-five global regression gives intercepts of 22.4 and 22.1. The new discontinuity estimate is:

\hat{\tau}_{global,new}=22.1-22.4=-0.3\text{ outcome units}.

Thus the estimated global jump changes by \(-0.3-2.4=-2.7\) outcome units after that one outcome is increased by 8. The change is consistent with the least-squares sensitivity formula: a change in one outcome changes the estimated coefficient vector, including the intercept. If the observation lies outside a local fit’s chosen neighborhood, that fit is unchanged by this particular perturbation. This does not mean distant observations should automatically be deleted; it means their influence on a global cutoff estimate should be understood.

Worked Example: Uneven Support and a Misleading Global Curve

In a third hypothetical simulation, 900 units have scores between \(-8\) and \(8\), but most scores lie between \(-2\) and \(2\); only a few observations occur in either tail. The simulated outcome jump is 1.25 units. A local linear fit within two score units estimates the left and right limits as 30.7 and 31.8. A local quadratic fit over the same neighborhood estimates them as 30.5 and 31.8.

The local estimates are obtained by subtracting the fitted left limit from the fitted right limit:

\begin{aligned} \hat{\tau}_{local,linear}&=31.8-30.7=1.1\text{ outcome units},\\ \hat{\tau}_{local,quadratic}&=31.8-30.5=1.3\text{ outcome units}. \end{aligned}

A separate degree-six global fit across the full range yields intercepts of 29.4 and 33.6. Its estimate is:

\hat{\tau}_{global}=33.6-29.4=4.2\text{ outcome units}.

Here the local linear and quadratic estimates are near the simulated jump, while the global estimate is much larger. The sparse tails make the global shape less securely supported at the edges, and the high-degree fit can bend while trying to accommodate patterns across the full range. One simulation cannot establish that a particular local specification is generally preferable, but it shows why a large global-polynomial estimate should not be accepted merely because the polynomial fits the observed outcomes.

What to Examine in Practice

A responsible RD analysis should make the relationship between the estimation method and the local estimand clear. The earlier material on bandwidth choice explains the bias-variance tradeoff involved in selecting the neighborhood for local estimation. The present issue is different: a global high-degree fit may use observations far from the cutoff to shape the estimated limits, even when the causal question concerns behavior at the cutoff.

When assessing a proposed high-order global fit, inspect the fitted curves on each side, paying particular attention to the region close to zero and the boundaries of the score range. Look for sharp bends, oscillations, and areas where the fit is weakly supported by observed scores. Compare the cutoff estimate with a local polynomial estimate whose fitting neighborhood is stated, and check whether the global estimate changes markedly when the polynomial degree, score range, or influential observations change. These comparisons are diagnostics; none alone proves identification or establishes that one estimate is unbiased.

Common mistake: “The high-degree polynomial fits the whole score range more closely, so its RD estimate must be better.” A closer global fit can reflect extra flexibility, including unstable bending that does not improve estimation at the cutoff. The correction is to focus on the cutoff limits, inspect support and curve shape near the threshold, and assess sensitivity to the global specification. A good overall fit is not, by itself, evidence of a credible local discontinuity estimate.

Nor should the opposite claim be made: local polynomial estimation is not automatically reliable simply because it uses nearby observations. Its neighborhood, polynomial order, support, and uncertainty all matter. The key lesson is narrower and practical: a high-order global polynomial is not a default solution to functional-form concerns in RD. Its flexibility can create oscillation and sensitivity exactly where the discontinuity is estimated.

Check Your Understanding

Answer each question by referring to the cutoff estimand and the role of the fitted curve.

In a separate degree-\(p\) global polynomial RD fit, which two coefficients determine the estimated discontinuity at a cutoff normalized to zero?
Why can an observation far from the cutoff affect a global polynomial estimate of the jump?
Give two distinct warning signs of instability in a high-order global RD fit.
A local fit estimates limits of 18.6 and 19.1 outcome units, while a global fit estimates intercepts of 17.0 and 20.4. Calculate both estimated jumps and their difference.
Why does a closer fit to outcomes across the entire score range not establish that a global polynomial provides a better estimate at the cutoff?